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2018 | Book

Basic Chemical Graph Theory Read chapter Read first chapter

Multi-shell Polyhedral Clusters

Author: Prof. Dr. Mircea Vasile Diudea

Publisher: Springer International Publishing

Book Series : Carbon Materials: Chemistry and Physics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This volume presents new methodologies and rationalizes existing methods that are used in the design of multi-shell polyhedral clusters. The author describes how the methods used are extended from 2D-operations on maps to 3D (and higher dimensional) Euclidean space. A variety of structures is designed and described in detail and classified giving rise to an atlas of multi-shell nanostructures. The book therefore sheds a new light on the field of crystal and quasicrystal structures, an important part of nanoscience and nanotechnology. The author goes on to show how the recently established methods are used for building complex multi-shell nanostructures and how this completes the existing information in the field. The atlas of such structures is completed with atomic coordinates (included as supplementary material). The content of this book gives a useful insight into structure elucidation and suggests new material synthesis.

Table of Contents

Frontmatter
Chapter 1. Basic Chemical Graph Theory
Abstract
Graph Theory applied in Chemistry is called Chemical Graph Theory. This interdisciplinary science takes problems (like isomer enumeration, structure elucidation, etc.) from Chemistry and solve them by Mathematics (using tools from Graph Theory, Set Theory or Combinatorics), thus influencing both Chemistry and Mathematics. This chapter introduces to basic definitions in Graph Theory: graph, walk, path, circuit, planar graph, graph invariant, vertex degree, chemical graph, etc. Then topological matrices are introduced: adjacency, distance, detour, combinatorial matrices, Wiener and Cluj matrices, walk matrix operator (combining three square matrices), reciprocal distance, and layer/shell matrices, on which the centrality indices are defined. Some info about topological symmetry is also presented.
Mircea Vasile Diudea
Chapter 2. Operations on Maps
Abstract
Design of structures discussed in this book is based on “operations on maps” (topological-geometrical modifications of a parent map), merely applied on the Platonic solids: tetrahedron (T), cube (C), octahedron (O), dodecahedron (D) and icosahedron (I); a map M is a discretized surface domain. The operations discussed in the Chap. 2 are: dual d, medial m, truncation t, polygonal mapping p n , snub s, leapfrog l, quadrupling/chamfering q, and septupling/wirl s n .
The figure sequence of a modified polyhedron: {v, e, f}, is given function of the edges e of the parent polyhedron; all the parameters herein presented refer to regular maps. The symmetry of parents is preserved by running these operations. An Atlas of single-shell clusters derived from the Platonic solids is also presented.
Mircea Vasile Diudea
Chapter 3. Definitions in Polytopes
Abstract
Multi-shell clusters represent complex structures, the study of which needs rigorous definitions in graph theory, geometry, set theory, etc. Within this chapter, main definitions for polyhedra, regular (Platonic) polyhedra, semi-regular and uniform (Archimedean, Catalan, Johnson’s) polyhedra are given. Then, higher dimensional polytopes are introduced, basically the regular polytopes. Euler formula for polyhedra, and then the alternating sum of higher ranked facets are used to confirm an assumed structure. Abstract polytopes, posets (replacing the dimension concept with that of rank), Hässe diagrams are also discussed. Polytope realization is exemplified by P-centered clusters and “cell-in-cell” clusters, as the simplest 4-dimensional/ranked structures.
Mircea Vasile Diudea
Chapter 4. Symmetry and Complexity
Abstract
Classical geometric symmetry refers to some operations acting on geometric properties of a polyhedron, that leave the object invariant; it is reflected in several molecular properties, such as dipole moments, IR vibrations, 13C-NMR signals etc. Topological symmetry, defined in terms of connectivity, is addressed to constitutive aspects of a molecule and it is involved in synthesis and/or structure elucidation. Complexity refers to the state or quality of being complex/intricate/complicated, or being the union of some interacting (by some rules) parts. Structural complexity is addressed to the organization of matter. It is studied by the aid of graphs associated to molecules/ions/crystals, on which basis several descriptors are calculated. Topological symmetry speaks about structural complexity by considering the type of atoms/vertices and their reciprocal distribution. Genus and rank (or space dimension) of a structure are parameters of complexity acting by means of Euler characteristic of the embedding surface. The fourth chapter introduces to: Euler characteristic, topological symmetry, indices of centrality, ring signature index and Euler characteristic, as reflected in pairs of map operations.
Mircea Vasile Diudea
Chapter 5. Small Icosahedral Clusters
Abstract
In geometry, any polyhedron with 12 faces is named a dodecahedron, among which only one is the regular dodecahedron (i.e. the Platonic solid), composed of 12 regular pentagonal faces, 3 of which meeting at each vertex; it has the Schläfli symbol {5,3} and icosahedral (point group) symmetry, I h . The dual of a dodecahedron is an icosahedron, referring to shapes, if one disregards the angles and bond length, rather than to regular polyhedra. The fifth chapter shows the transforming, by map operations, of small seeds, like “point centered polyhedra” and “cell-in-cell”, into more complex multi-shell clusters, of rank 4 or 5. Among the transformed polyhedra, a special attention was given to rhombic polyhedra, obtained by the sequence d(m(P)) (i.e. dual of medial polyhedra). An atlas section illustrates the discussed multi-shell polyhedral clusters.
Mircea Vasile Diudea
Chapter 6. Large Icosahedral Clusters
Abstract
Large multi-shell clusters, of icosahedral symmetry, are derived, by operations on maps, from seeds like Bergman’s C45 and a larger C125 clusters. The most used operations were medial and truncation, eventually followed by dualization. The clusters were characterized by figure count while their topological symmetry was described in terms of ring signature and centrality index. Clusters composed of dodecahedral shapes and having dodecahedral topology, both spongy and filled, were designed by an original procedure. Rhomb decorated clusters were realized by medial operation followed by dualization, or by p 4 operation. Attention was focused on the cluster C152 that includes the smallest rhombic Rh3 substructure (the skeleton of a real molecule, named [1,1,1]-propellane), which is not a polyhedron but a tile; a new class of structures, called “propellanes” was thus discovered. An atlas section illustrates the discussed multi-shell polyhedral clusters.
Mircea Vasile Diudea
Chapter 7. Clusters of Octahedral Symmetry
Abstract
Cube and its dual Octahedron exist in any multi-dimensional space; as shapes, they compose multi-shell clusters of octahedral symmetry (resulted by operations on maps) and crystal networks. A particular attention was given to clusters decorated with octahedra and dodecahedra, respectively.
Cube is the only Platonic solid that can tessellate the 3D space. This chapter was focused on two space fillers: the cube C and the rhombic dodecahedron, Rh12 (i.e. d(mC).14, or dual of cuboctahedron) and to their networks, derived by map operations, like dual, medial, truncation or leapfrog. The clusters and networks were characterized by figure count, ring signature and centrality index. An atlas section illustrates the discussed multi-shell polyhedral clusters and triple periodic structures, respectively.
Mircea Vasile Diudea
Chapter 8. Tetrahedral Clusters
Abstract
The first Platonic solid, Tetrahedron, is self-dual; higher dimensional analogues are called simplex/simplices; tetrahedral shapes can be found in vary polyhedral clusters. Adamantane-like structure, Ada20, is a hyper-tetrahedron, a tetrahedron of which points were changed by four tetrahedral units P@4C20; the central hollow has the topology of small fullerene C28; Ada20 is the unit of “diamond D5”, or MTN zeolite. Map operations, like medial m, truncation t and leapfrog l were applied to Ada20, to obtain a variety of spongy or filled structures. Tetrahedral hyper-structures decorated only with dodecahedra were also described. Figure count was used for characterization of the discussed clusters. An atlas section illustrates the discussed multi-shell polyhedral clusters.
Mircea Vasile Diudea
Chapter 9. C60 Related Clusters
Abstract
A hyper-structure is a (molecular) construction, of formula MY(mS; p) . n: M is the parent molecule in the Y “hyper”-building while m is the number of substructures S, joined by the type p facets; the total number n of atoms suffixes the name of the hyper-structure. Point-centered clusters and “cell-in-cell” clusters, used as seeds for map operations or even stellated “monomers” contributed to hyper-structure building, all based on the topology of M = C60(I h ). Examples are given starting from C750 = C60Y(60C20).750, transformed by dual, medial and truncation operations, to obtain hyper-structures of the type C60Y(60S; p).n.
Vertex equivalence classes found by topological descriptors were confirmed by permutations performed on the adjacency matrix associated to such complex graphs (by using Mathematica software). An atlas section illustrates the discussed multi-shell polyhedral clusters.
Mircea Vasile Diudea
Chapter 10. Chiral Multi-tori
Abstract
Chirality is one of the basic characteristics of biological structures; chirality is a symmetry property. Multi-tori are complex structures consisting of more than one torus, embedded in negatively curved surfaces. Design of multi-tori may be achieved by operations on maps. The “monomers” used to design the chiral multi-tori discussed in this chapter are snubs of Platonic solids. A snub polyhedron s(P) is achieved by dualizing the p 5(P) transform: s(P) = d(p 5(P)); since p 5-operation is prochiral, all the consecutive structures will be chiral; high genus structures of rank k = 3 were thus obtained. The genus and rank (or space dimension) of a structure are parameters of complexity.
Topological symmetry of the structures herein discussed was evaluated by ring signature and centrality index and confirmed by symmetry calculation using the adjacency matrix permutations. C60-related chiral tori were also designed and their symmetry evaluated. An atlas section illustrates the discussed chiral multi-tori.
Mircea Vasile Diudea
Chapter 11. Spongy Hypercubes
Abstract
Hypercube Q n is an n-dimensional analogue of the Cube (n = 3); it is a regular graph of degree n and can be obtained by the Cartesian product of P 2 graph or can be drawn as a Hӓsse diagram. Hypercube is a regular polytope in the space of any number of dimensions; its Schläfli symbols is {4,3 n−2} and has as a dual the n-orthoplex {3 n−2,4}. The number of k-cubes contained in an n-cube Q n (k) comes from the coefficients of (2k + 1) n . A “spongy hypercube” G(d, v, Q n + 1) = G(d, v) □ n P 2 is defined in this chapter; on each edge of the original polyhedral graph, a local hypercube Q n is evolved; these hypercubes are incident in a hypervertex, according to the original degree, d. It means that, in a spongy hypercube, the original 2-faces are not counted.
The k-faces of a spongy hypercube are combinatorially counted from the previous rank faces; their alternating summation accounts for the genus of the embedded surface. Tubular and toroidal hypercubes were also designed. Analytical formulas for counting Omega and Cluj polynomials, respectively, in hypercubes were derived.
Mircea Vasile Diudea
Chapter 12. Energetics of Multi-shell Clusters
Abstract
Multi-shell clusters may be viewed as realizations of abstract structures, representing ways of the space filling, either in compact or spongy manner, by cells representing shapes of the geometrical bodies; such structures refer rather to crystal/quasi-crystal state than to molecules. This chapter brings some computational arguments in the favor of (carbon) nanostructures described within the book. Aggregation of C20 shapes within the D5 diamond, with adamantane-like “Ada”, diamantane-like “Dia”, and fivefold stars substructures were designed and computed at DFTB level of theory. Hyper-graphenes derived from the D5 substructures were also considered. Analogously, C60-based hyper-graphenes were designed and substructures computed at DFTB, HF and DFT levels of theory. Aggregation of C60 in clusters of tetrahedral or icosahedral symmetry were designed and computed at DFT or MP6 levels of theory. Networks with C60[2+2] cycloadducts, in several topologies were also computed. An atlas section illustrates the discussed multi-shell polyhedral clusters and crystal networks.
Mircea Vasile Diudea
Backmatter
Metadata
Title
Multi-shell Polyhedral Clusters
Author
Prof. Dr. Mircea Vasile Diudea
Copyright Year
2018
Electronic ISBN
978-3-319-64123-2
Print ISBN
978-3-319-64121-8
DOI
https://doi.org/10.1007/978-3-319-64123-2

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